1.1 It might be supposed that I had chosen the title of this address to fit in with the great national debate on education which has taken place since the Prime Minister’s speech at Ruskin College in October last. However, this is not so (and thus this is not a trendy genuflexion to these recent events). It is entirely fortuitous that the theme does enable me to make references to the debate and to some of the issues raised by it. Before and at the Conference last year I kept wondering what approach I should take in this address. I knew that I wished to choose a theme which fitted in with my personal and recent experiences.

It has now been possible to study the many comments received since the publication of the “interim statement” in Gazette No. 407 (March 1975). The recommendations given below are based on the original suggestions, taking these comments into account. Sidelining indicates a change or modification of the original suggestion.

It is proposed also to publish these recommendations as a pamphlet, so that they may be widely available to authors and others.

Popular expositions of the new subject of catastrophe theory have tended to concentrate on ‘soft’ applications, but, as I hope to show, it is perfectly possible to make a serious, important, and mathematically more demanding application intelligible to sixth formers with a minimum of differential calculus. The application is in that part of engineering science which deals with the stability of structures. Catastrophe theory produces nothing really novel in this field, but, by assuming Thorn’s theorem, which will be explained below, it is possible to make the whole subject accessible to those for whom Taylor’s theorem for a number of variables, and the associated concepts, are too difficult. Even for engineers already familiar with the results, catastrophe theory offers an attractive unity within the subject and can point to analogies in related fields.

It is just twenty years ago that sixty school teachers (mostly heads of mathematics departments) and twenty representatives of industry spent ten days together at Trinity College, Oxford for an intensive exchange of information and ideas. They had as their theme the changes that had been taking place in the uses of mathematics, and their relevance for school teaching. The conference was a seminal one in the development of British mathematical education; it made important personal contacts, it widened horizons, and it could also be described as the great-grandfather of SMP (the grandfather and father being the conferences which followed in 1959 and 1961 at Liverpool and Southampton). With the emphasis nowadays on the importance of keeping school mathematics in touch with the needs of industry, it is germane to remind ourselves of the influence which industrialists have exercised—personally, financially and mathematically—on the development and dissemination of new curricula in this country. Members of our Association may not always appreciate the contribution of its Schools and Industry Committee in keeping the dialogue going.

Many school textbooks state, but few of them prove, that the determinant of a 2 × 2 matrix is the area scale factor of its transformation; whilst a proof of the corresponding result for 3 × 3 matrices is usually delayed until the scalar triple product is available as a technique. Teachers of SMP, or of other ‘modern’ courses containing some development of transformation geometry, may be interested in the following approach to the topic of determinants.

A few years ago, a well known laundry detergent manufacturer ran a sales-promotion competition in Australia called “Jack-in-the-box”, with the following rules:

1. 1. Draw a continuous line through 13 of the numbered squares (see Fig. 1). The numbers in the squares through which the line has been drawn are then added to give a total score. Entries achieving the highest total scores will qualify for one of the prizes.

2. 2. The line must not go through any square more than once. It also cannot be drawn diagonally through the corner of any squares_only up and down or across (see the example in Fig. 1, for which the score is 723).

3. 3. The line must begin at the square J (for Jack-in-the-box) and may finish in any square. The J square does not count as one of the 13 squares.

For some mechanical puzzles there exists a complete mathematical analysis. The purpose of this note is to describe one such puzzle, together with a number of variations of it. In the classroom the puzzle could be used as a teaching aid to introduce children to binary notation, mathematical induction, congruences, as well as some of the concepts of group theory. My own ten-year-old son was delighted to find that counting in binary notation, an idea to which he had been introduced in a purely theoretical way, leads to an easy method of solving the puzzle.

A teacher can make a great deal out of some of the questions which pupils and friends put to him from time to time. In an article in Mathematics in School a few years ago [1] I listed a number of such questions and discussed one, which involved a geographical application of matrices, in detail. The present article continues this theme, but its main aim is to stimulate discussion of the place of “mathematical modelling” in the curriculum. For it is felt by some that, in spite of the opportunities offered by (say) linear programming, mechanics and probability, insufficient emphasis is placed on modelling in the secondary school. It has been said too that we have not been notably successful in finding good applications of topics new to our syllabuses and incorporating them into our teaching [2].

There seems little doubt that the proportion of the mathematics sixth form who include traditional applied mathematics as part of their studies is steadily decreasing, and that this decline in its turn affects both the popularity of, and the ability of students to cope with, applied mathematics in universities. It is evident that fewer sixth formers are taking the ‘double mathematics plus physics’ route to reading mathematics at degree level; the combinations taking its place are many and varied, but often include statistics, for example, instead of applied mathematics, and economics or geography or a language instead of physics. Even where some students take so-called double mathematics, the syllabus followed often gives statistics as an alternative to applied mathematics, and an increasing number of schools are teaching this option.

When I began teaching secondary mathematics after several years in industry, I hadn’t been trained as a teacher. Having been away from school mathematics for so long, I was sometimes nonplussed by pupils’ questions.

One day I was showing a mixed-ability first-year class how to obtain the decimal expansion of a vulgar fraction. After explaining the how and, I hoped, the why of the method I did several examples on the blackboard and concluded by telling them that the expansion would either terminate or recur (or words to that effect). At once a small boy demanded to be told how I knew this. This floored me: I hadn’t got a ready answer and I had to promise to think about it.

A good deal of interesting mathematics at various levels arises from the situation where a loan is repaid by regular equal instalments, with interest being charged on the balance outstanding at each stage. The loan may be a mortgage, in which case the instalments are reckoned as annual (even though they are actually paid monthly), or a hire purchase debt or a personal loan of some sort with interest reckoned monthly. The following topics are in roughly increasing order of difficulty.

The problem discussed in this article is to find the number of patterns that can be formed by colouring the faces of a cube, each face to be of one colour, colourings being considered to be of the same pattern if they can be made to coincide by rotation of the cube. To make the method clear, we shall begin by considering the analogous problem of colouring a hexagon divided into six equilateral triangles.

Apart from the conic sections, many well known curves have physical and geometrical applications which are readily understood by sixth formers and undergraduates. The catenary, cycloid, spirals, involutes, evolutes and many others appear in courses in civil engineering and control mechanisms, and frequently in A level calculus questions.

A tiling of the plane is a family of sets, called tiles, that cover the plane without gaps or overlaps. Usually we are concerned with tilings whose tiles are of a small number of different shapes; familiar examples are the regular and uniform tilings (see, for example, [1]).

Although the mathematical theory of tiling is very old, it still contains a rich supply of interesting and challenging problems. The purpose of this article is to describe some of these. In many cases the mathematical aspect is enhanced by the aesthetic appeal of the resulting tilings.

1. Introduction. Recently I was looking for a simple but interesting model which a class of students could complete in about twenty minutes, which would illustrate groups of symmetries in three-dimensional space. My thoughts went back to the lampshade described in the Gazette for June 1971 (No. 393, pp. 310–312) but I felt that it would take too long to make, even if each student merely cut out a strip which I had marked out beforehand. Forced in this way to reconsider the matter, I found that the last sentence of my article was false—there are in fact four other such structures, which I now describe.

Mathematics has had a profound influence on economic theory within the last 25 years, and many well known economists started their higher education in mathematics. Alfred Marshall and John Maynard Keynes, for example, were both Wranglers at Cambridge; we also know that Adam Smith—whose Inquiry into the nature and causes of the wealth of nations was published just over 200 years ago—studied logic and mathematics at Glasgow; and, more recently, some scholars have come to regard Karl Marx as one of the ‘giants’ in the history of mathematical economics.

Among the definitions at the beginning of Book I in Euclid’s Elements [1] there are several that pick out special kinds of triangles and quadrilaterals. In his commentary [2] on Book I, Proclus observes that Euclid classifies triangles in two ways: firstly ‘by sides’ into equilateral, isosceles and scalene triangles; and secondly ‘by angles’ into right-angled, obtuse-angled and acute-angled triangles. With regard to quadrilaterals, Proclus ([2], p. 134) attributes to Posidonius the classification scheme on p. 39, which is to be found in Heath’s edition of the Elements ([1], p. 189). Thus the ancient classification of triangles and quadrilaterals produces three (or six) species of triangles and seven species of quadrilaterals.

Dr T. J. Fletcher’s article in last December’s Gazette [1] was of great interest mathematically in illustrating how an eminently practical problem (here, essentially a combinatorial problem) could be solved using some surprising and unexpectedly theoretical mathematical ideas. (These included the field GF(22), the group (C2)4 and finite geometries, as well as excursions into Pappus’ and Desargues’ theorems.) But to myself, at least, the interest was augmented because the article provided the background for three or four end-of-term lessons with a third year grammar school form. The purpose of this article is to describe how some of the material was adapted.